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Background
• We Have Discussed the Construction and Estimation of DSGE Models• Next, We Turn to Analysis• Most Basic Policy Question:
– How Should the Policy Variables of the Government be Set?– What is Optimal Policy? What Should R Be, How Volatile Should P Be?
• In Past 10 Years, Profession Has Explored Operating Characteristics of SimplePolicy Rules– One Finding: A Taylor Rule with High Weight on Inflation Works Well in
New-Keynesian Models• Recent Development:
– Increasingly, Analysts Studying Optimal Policy– Perhaps Because there is a Perception that Current DSGE Models Fit Data
Well• We Will Review Some of this Work.
10
Modern Quantitative Analysis of Optimal Policy
• Case Where Intertemporal Government Budget Constraint Does Not Bind
– Example - Current Generation of Monetary Models∗ Assume Presence of Lump-Sum Taxes Used to Ensure Government
Budget Constraint is Satisfied
– Optimal Policy Studied, Among Others, By Schmitt-Grohe and Uribe(2004), Levin, Onatski, Williams, Williams (2005), and References TheyCite.
• Case Where Intertemporal Government Budget Constraint Binds
– Example - When the Government Does not Have Access to Distorting Taxes– Chari-Christiano-Kehoe (1991, 1994), Schmitt-Grohe and Uribe (2001),
Siu (2001), Benigno-Woodford (2003, 2005), Others.
13
Outline
• Optimal Monetary and Fiscal Policy When the Intertemporal Budget ConstraintBinds
– Analyze the Friedman-Phelps Debate over the Optimal Nominal Rate ofInterest.
– What is the Optimal Degree of Price Variability?– How Should Policy React to a Sudden Jump in G?– Log-Linearization as a Solution Strategy– Woodford’s Timeless Perspective
• Optimal Monetary Policy When the Intertemporal Budget Constraint Can beIgnored.
– Log-Linearization as a Solution Strategy
14
Optimal Policy in the Presence of a BudgetConstraint
• Sketch of Phelps-Friedman Debate
• Some Ideas from Public Finance - Primal Problem
• Simple One-Period Example
• Determining Who is Right, Friedman or Phelps, Using Lucas-Stokey Cash-Credit Good Model
• Financing a Sudden Expenditure (Natural Disaster): Barro versus Ramsey.
15
Friedman-Phelps Debate
• Money Demand:M
P= exp[−αR]
• Friedman:
a. Efforts to Economize Cash Balances when R High are Socially Wasteful
b. Set R as Low As Possible: R = 1.
c. Since R = 1 + r + π, Friedman Recommends π = −r.
i. r ∼ exogenous (net) real interest rate rateii. π ∼ inflation rate, π = (P − P−1)/P−1
16
Friedman-Phelps Debate ...
• Phelps:a. Inflation Acts Like a Tax on Cash Balances -
Seigniorage =Mt −Mt−1
Pt=Mt
Pt− Pt−1
Pt
Mt−1Pt−1
≈ M
P
π
1 + π
b. Use of Inflation Tax Permits Reducing Some Other Tax Rate
c. Extra Distortion in Economizing Cash Balances Compensated by ReducedDistortion Elsewhere.
d. With Distortions a Convex Function of Tax Rates, Would Always Want toTax All Goods (Including Money) At Least A Little.
e. Inflation Tax Particularly Attractive if Interest Elasticity of Money DemandLow.
17
Question: Who is Right, Friedman or Phelps?
• Answer: Friedman Right Surprisingly Often
• Depends on Income Elasticity of Demand for Money
• Will Address the Issue From a Straight Public Finance Perspective, In theSpirit of Phelps.
• Easy to Develop an Answer, Exploiting a Basic Insight From Public Finance.
18
Question: Who is Right, Friedman or Phelps? ...
Some Basic Ideas from Ramsey Theory• Policy, π, Belonging to the Set of ‘Budget Feasible’ Policies, A.• Private Sector Equilibrium Allocations, Equilibrium Allocations, x,
Associated with a Given π; x ∈ B.
• Private Sector Allocation Rule, mapping from π to x (i.e., π : A→ B).• Ramsey Problem: Maximize, w.r.t. π, U(x(π)).• Ramsey Equilibrium: π∗ ∈ A and x∗, such that π∗ solves Ramsey Problem
and x∗ = x(π∗). ‘Best Private Sector Equilibrium’.• Ramsey Allocation Problem: Solve, x = argmax U(x) for x ∈ B
• Alternative Strategy for Solving the Ramsey Problem:a. Solve Ramsey Allocation Problem, to Find x.b. Execute the Inverse Mapping, π = x−1(x).
c. π and x Represent a Ramsey Equilibrium.
• Implementability Constraint: Equations that Summarize Restrictions onAchievable Allocations, B, Due to Distortionary Tax System.
19
Question: Who is Right, Friedman or Phelps? ...
Policy, π
Set, A, of Budget-Feasible Policies
Private sector Allocation Rule, x(π)
Set, B, of Private Sector Allocations Achievable by Some Budget-Feasible Policy
Private Sector Equilibrium
Allocations, x
Utility
20
Example ...
• Household Problem Implies Private Sector Allocation Rules, l(τ ), c(τ ),defined by:
ucz(1− l) + ul = 0, c = (1− τ )zl
l(τ) l
u[z(1-τ)l,l] ucz(1-τ)+ul=0
Private Sector Allocation Rules: l(τ), c(τ) = z(1-τ)l
22
Example ...
• Ramsey Problem:
maxτ
u(c(τ ), l(τ ))
subject to g ≤ zl(τ )τ
• Ramsey Equilibrium: τ ∗, c∗, l∗ such that
a. c∗ = c(τ ∗), l∗ = l(τ ∗)
∗ ‘Private Sector Allocations are a Private Sector Equilibrium’
b. τ ∗ Solves Ramsey Problem
∗ ‘Best Private Sector Equilibrium’
23
Analysis of Ramsey Equilibrium
• Simple Utility Specification:
u(c, l) = c− 12l2
• Two Ways to Compute the Ramsey Equilibrium
a. Direct Way: Solve Ramsey Problem (In Practice, Hard)
b. Indirect Way: Solve Ramsey Allocation Problem, or Primal Problem (CanBe Easy)
24
Analysis of Ramsey Equilibrium ...
Direct Approach• Private Sector Allocation Rules:
c(τ ) = z2(1− τ )2, l(τ ) = z(1− τ )
• ‘Utility Function’ for Ramsey Problem:
u(c(τ ), l(τ )) =1
2z2(1− τ )2
• Constraint on Ramsey Problem:g ≤ zl(τ )τ = z2(1− τ )τ
• Ramsey Problem:maxτ
1
2z2(1− τ )2
subject to : g ≤ τz2(1− τ ).
34
Analysis of Ramsey Equilibrium ...
¼z2
g
τz2(1-τ) ‘Laffer Curve’
0 1 τ
½z2(1-τ)2
τ1 τ2
τ* = τ1 =½ -½[ 1 – 4 g/z2 ]½ τ2 =½+½[ 1 – 4 g/z2 ]½
l(τ*) =½{ z+[ z2 – 4 g ]½ }
A
Government Preferences
35
Analysis of Ramsey Equilibrium ...
Indirect Approach• Approach: Solve Ramsey Allocation Problem, Then ‘Inverse Map’ Back into
Policies• Problem: Would Like a Characterization of B that Only Has (c, l), Not the
PoliciesB = {c, l : ∃τ , with ucz(1− τ ) + ul = 0,
c = (1− τ )zl, g ≤ τzl}• Solution: Rearrange Equations in B, So That Only (c, l) Appears
(∗) ucc + ull = 0, (∗∗) c + g ≤ zl.
• Conclude: B = D, where:
D =
⎧⎪⎪⎨⎪⎪⎩(c, l) : c + g ≤ zl| {z }resource constraint
, ucc + ull = 0| {z }implementability constraint
⎫⎪⎪⎬⎪⎪⎭
43
Analysis of Ramsey Equilibrium ...
• Express Ramsey Allocation Problem:
maxc,l
u(c, l), subject to (c, l) ∈ D
• Alternatively:
maxc,l
u(c, l),
s.t. ucc + ull = 0, c + g ≤ zl
• Or,
maxl
1
2l2
s.t. l2 + g ≤ zl
46
Analysis of Ramsey Equilibrium ...
0
Ramsey Allocation Problem:
Max ½l2 Subject to l2 + g ≤ zl Solution: l2 = ½{ z + [ z2 - 4g ]½ } Same Result as Before!
l2 - zl +g = 0
½z l1 l2
g - ¼z2
½l2
47
Analysis of Ramsey Equilibrium ...
Lucas-Stokey Cash-Credit Good Model• Households• Firms• Government
48
Analysis of Ramsey Equilibrium ...
Households• Household Preferences:
∞Xt=0
βtu(c1t, c2t, lt),
c1t ˜ cash goods, c2t ˜ credit goods, lt ˜ labor
• Distinction Between Cash and Credit Goods:
– All Goods Paid With Cash At the Same Time, After Goods Market, in AssetMarket
– Cash Good: Must Carry Cash In Pocket Before Consuming It
Mt ≥ Ptc1t
– Credit Good: No Need to Carry Cash Before Purchase.
49
Analysis of Ramsey Equilibrium ...
Household Participation in Asset and Good Markets
• Asset Market: First Half of Period, When Household Settles Financial ClaimsArising From Activities in Previous Asset Market and in Previous GoodsMarket.
• Goods Market: Second Half of Period, Goods are Consumed, Labor Effort isApplied, Production Occurs.
50
Analysis of Ramsey Equilibrium ...
t t+1
Asset Market Goods Market
Sources of Cash for Household: • Md
t-1 - Pt-1c1,t-1 - Pt-1c2,t-1 • Rt-1Bd
t-1 • (1-τt-1)zlt-1
Uses of Cash • Bonds, Bd
t • Cash, Md
t
• c1,t , c2,t Purchased • lt Supplied • Production Occurs • Md
t Not Less Than Ptc1,t
• Constraint On Households in Asset Market (Budget Constraint)
Mdt +Bd
t
≤ Mdt−1 − Pt−1c1t−1 − Pt−1c2t−1
+Rt−1Bdt−1 + (1− τ t−1)zlt−1
51
Analysis of Ramsey Equilibrium ...
Household First Order Conditions• Cash versus Credit Goods:
u1tu2t= Rt
• Cash Goods Today versus Cash Goods Tomorrow:
u1t = βu1t+1RtPt
Pt+1
• Credit Goods versus Leisure:
u3t + (1− τ t)zu2t = 0.
52
Analysis of Ramsey Equilibrium ...
Firms• Technology: y = zl
• Competition Guarantees Real Wage = z.
53
Analysis of Ramsey Equilibrium ...
Government• Inflows and Outflows in Asset Market (Budget Constraint):
Mst −Ms
t−1 +Bst| {z }
Sources of Funds
≥ Rt−1Bst−1 + Pt−1gt−1 − Pt−1τ t−1zlt−1| {z }
Uses of Funds
• Policy:
π = (Ms0 ,M
s1 , ..., B
s0, B
s1, ..., τ 0, τ 1, ...)
54
Analysis of Ramsey Equilibrium ...
Ramsey Equilibrium• Private Sector Allocation Rule:
For each policy, π ∈ A, there is a Private Sector Equilibrium:x = ({c1t} , {c2t} , {lt} , {Mt} , {Bt})
p = ({Pt} , {Rt})Mt =Ms
t =Mdt
Bt = Bst = Bd
t
Rt ≥ 1 (i.e., u1t/u2t ≥ 1)• Ramsey Problem:
maxπ∈A
U(x(π))
• Ramsey Equilibrium:π∗, x(π∗), p(π∗),
Such that π∗ Solves Ramsey Problem.
55
Finding The Ramsey Equilibrium By Solving theRamsey Allocation Problem
max{c1t,c2t,lt}∈D
∞Xt=0
βtu(c1t, c2t, lt),
where D is the set of allocations, c1t, c2t, lt, t = 0, 1, 2, ..., such that
∞Xt=0
βt[u1tc1t + u2tc2t + u3tlt] = u2,0a0,
c1t + c2t + g ≤ zlt,u1tu2t≥ 1,
a0 =R−1B−1
P0∼ real value of initial government debt
56
Lagrangian Representation of Ramsey AllocationProblem
• There is a λ ≥ 0, s. t. Solution to R A Problem Also Solves:
max{c1t,c2t,lt}
∞Xt=0
βtu(c1t, c2t, lt) + λ
à ∞Xt=0
βt[u1tc1t + u2tc2t + u3tlt]− u2,0a0
!subject to c1t + c2t + g ≤ zlt,
u1tu2t≥ 1,
or,
max{c1t,c2t,lt}
W (c10, c20, l0;λ) +∞Xt=1
βtWt(c1t, c2t, lt;λ)
subject to : c1t + c2t + g ≤ zlt,u1tu2t≥ 1,
W (c10, c20, l0;λ) = u(c1,0, c2,0, l0) + λ ([u1,0c1,0 + u2,0c2,0 + u3,0l0]− u2,0a0)
W (c1,t, c2,t, lt;λ) = u(c1,t, c2,t, lt) + λ ([u1,tc1,t + u2,tc2,t + u3,tlt)
58
Ramsey Allocation Problem
• Lagrangian:
max{c1t,c2t,lt}
W (c10, c20, l0;λ) +∞Xt=1
βtW (c1t, c2t, lt;λ)
subject to : c1t + c2t + g ≤ zlt,u1tu2t≥ 1,
W (c10, c20, l0;λ) = u(c1,0, c2,0, l0) + λ ([u1,0c1,0 + u2,0c2,0 + u3,0l0]− u2,0a0)
W (c1,t, c2,t, lt;λ) = u(c1,t, c2,t, lt) + λ ([u1,tc1,t + u2,tc2,t + u3,tlt)
• How to Solve this?– Fix λ ≥ 0, Solve The Above Problem– Evaluate Implementability Constraint– Adjust λ Until Implemetability Constraint is Satisfied
61
Special Structure of Ramsey Allocation Problem
• Given λ (If we Ignore u1tu2t≥ 1), Looks Like Standard Optimization Problem:
max{c1t,c2t,lt}
W (c10, c20, l0;λ) +∞Xt=1
βtW (c1t, c2t, lt;λ)
s.t. c1t + c2t + g ≤ zlt.
• After First Period, ‘Utility Function’ Constant
• Problem: For Exact Solution, Need λ...Not Easy to Compute!
• But,– Can Say Much Without Knowing Exact Value of λ (Will Pursue this Idea
Now)– Under Certain Conditions, Can Infer Value of λ From Data (Will Pursue
this Idea Later)
65
Special Structure of Ramsey Allocation Problem ...
• Ignoring u1tu2t≥ 1, after Period 1 :
W1(c1, c2, l;λ)
W2(c1, c2, l;λ)= 1
• ‘Planner’ Equates Marginal Rate of Substitution Between Cash and CreditGood to Associated Marginal Rate of Technical Substitution
66
Restricting the Utility Function
• Utility Function:
u(c1, c2, l) = h(c1, c2)v(l),
h ∼ homogeneous of degree k, v ∼ strictly decreasing.
• Then, u1c1 + u2c2 + u3l = h [kv + v0], so
W (c1, c2, l;λ) = hv + λh [kv + v0] = h(c1, c2)Q(l, λ).
• Conclude - Homogeneity and Separability Imply:
1 =W1(c1, c2, l;λ)
W2(c1, c2, l;λ)=h1(c1, c2, l)Q(l, λ)
h1(c1, c2, l)Q(l, λ)=h1(c1, c2, l)
h1(c1, c2, l)=u1(c1, c2, l)
u2(c1, c2, l).
70
Surprising Result: Friedman is Right More OftenThan You Might Expect
• Suppose You Can Ignore u1t/u2t ≥ 1 Constraint. Then, Necessary Conditionof Solution to Ramsey Allocation Problem:
W1(c1, c2, l;λ)
W2(c1, c2, l;λ)= 1.
• This, In Conjunction with Homogeneity and Separability, Implies:
u1(c1, c2, l)
u2(c1, c2, l)= 1.
• Note: u1t/u2t ≥ 1 is Satisfied, So Restriction is Redundant Under Homogene-ity and Separability.
• Conclude: R = 1, So Friedman Right!
72
Generality of the Result
• Result is True for the Following More General Class of Utility Functions:
u(c1, c2, l) = V (h(c1, c2), l),
where h is homothetic.
• Analogous Result Holds in ‘Money in Utility Function’ Models and ‘Transac-tions Cost’ Models (Chari-Christiano-Kehoe, Journal of Monetary Economics,1996.)
• Actually, strict homotheticity and separability are not necessary.
73
Interpretation of the Result
• ‘Looking Beyond the Monetary Veil’ -
– The Connection Between The R = 1 Result and the Uniform TaxationResult for Non-Monetary Economies
• The Importance of Homotheticity
– The Link Between Homotheticity and Separability, and The ConsumptionElasticity of Money Demand.
74
Uniform Taxation Result from Public Finance ForNon-Monetary Economies
• Households:
maxc1,c2,l
u(c1, c2, l) s.t. zl ≥ c1(1 + τ 1) + c2(1 + τ 2)
⇒ c1 = c1(τ 1, τ 2), c2 = c2(τ 1, τ 2), l = l(τ 1, τ 2).
• Ramsey Problem:
maxτ 1,τ 2
u(c1(τ 1, τ 2), c2(τ 1, τ 2), l(τ 1, τ 2))
s.t. g ≥ c1(τ 1, τ 2)τ 1 + c2(τ 1, τ 2)τ 2
• Uniform Taxation Result:if u = V (h(c1, c2), l), h ∼ homothetic
then τ 1 = τ 2.
Proof : trivial! (just study Ramsey Allocation Problem)
77
Similarities to Monetary Economy
• Rewrite Budget Constraint:
zl
1 + τ 2≥ c1
1 + τ 11 + τ 2
+ c2.
• Similarities:
1
1 + τ 2∼ 1− τ ,
1 + τ 11 + τ 2
∼ R.
• Positive Interest Rate ‘Looks’ Like a Differential Tax Rate on Cash and CreditGoods.
• Have the Same Ramsey Allocation Problem, Except Monetary EconomyAlso Has:
u1u2≥ 1.
79
What Happens if You Don’t Have Homotheticity?
• Utility Function:
u(c1, c2, l) =c1−σ1
1− σ+
c1−δ2
1− δ+ v(l)
• ‘Utility Function’ in Ramsey Allocation Problem:
W (c1, c2, l) = [1 + (1− σ)λ]c1−σ1
1− σ
+ [1 + (1− δ)λ]c1−δ2
1− δ+ v(l) + λv0(l)l
81
What Happens if You Don’t Have Homotheticity? ...
• Marginal Rate of Substitution in Ramsey Allocation Problem That Ignoresu1/u2 ≥ 1 Condition:
1 =W1(c1, c2, l;λ)
W2(c1, c2, l;λ)=1 + (1− σ)λ
1 + (1− δ)λ× u1u2,
or, since u1/u2 = R :
R =1 + (1− δ)λ
1 + (1− σ)λ• Finding:
δ = σ ⇒ R = 1 (homotheticity case)δ > σ ⇒ R ≥ 1 Binds, so R = 1δ < σ ⇒ R > 1.
Note: Friedman Right More Often Than Uniform Taxation Result, Becauseu1/u2 ≥ 1 is a Restriction on the Monetary Economy, Not the Barter Economy.
82
Consumption Elasticity of Demand
• Homotheticity and Separability Correspond to Unit Consumption Elasticity ofMoney Demand.
• Money Demand:
R =u1u2=h1h2= f
µc2c1
¶
= f
Ãc− M
PMP
!
= f
µc
M/P
¶.
• Note: Holding R Fixed, Doubling c Implies Doubling M/P
83
Money Demand and Failure of Homotheticity
• Money Demand:
R =u1u2=c−σ1c−δ2
=
¡MP
¢−σ¡c− M
P
¢−δ• Taylor Series Approximation About Steady State (m ≡M/P in steady state) :
m =1
mc +
σδ
¡1− m
c
¢| {z }×cConsumption Money Demand Elasticity, εM
− 1
δ mc−m + σ| {z }×RInterest Elasticity
• Can Verify:
Utility Function Non-Monetary MonetaryParameters εM Economy Economyδ > σ εM > 1 τ 2 ≥ τ 1 R = 1δ < σ εM < 1 τ 2 < τ 1 R > 1δ = σ εM = 1 τ 1 = τ 2 R = 1
84
Bottom Line:
• Friedman is Right (R = 1) When Consumption Elasticity of Money Demandis Unity or Greater
• Implicitly, High Interest Rates Tax Some Goods More Heavily that Others.Under Homotheticity and Separability Conditions, Want to Tax Goods at SameRate.
85
Bottom Line: ...
• What is Consumption Elasticity in the Data?
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 20050.5
1
1.5
Vel
ocity
date
Federal Funds Rate and Consumption Velocity of St. Louis Fed’s MZM
VelocityFederal Funds Rate
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 20050.5
1
1.5
Vel
ocity
date
Federal Funds Rate and Consumption Velocity of St. Louis Fed’s MZM
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 20050
0.1
0.2
Fed
eral
Fun
ds R
ate
Funds Rate
Velocity
• Answer: Not Far From Unity - Velocity and the Interest Rate Are BothRoughly Where they Were in the 1960, Though Consumption is Higher.
87
1980 1985 1990 1995 2000 2005 20100.7
0.8
0.9
1
1.1C
onsu
mpt
ion
Vel
ocity
of M
1Euro-area Velocity and Interest Rate
1980 1985 1990 1995 2000 2005 20100
5
10
15
20
Rat
e on
3-m
onth
Eur
ibor
, AP
R
1980 1985 1990 1995 2000 2005 20100.7
0.8
0.9
1
1.1
Con
sum
ptio
n V
eloc
ity o
f M1
Euro-area Velocity and Interest Rate
1980 1985 1990 1995 2000 2005 20100
1
2
3
4
Rat
e on
Ove
rnig
ht d
epos
its, A
PR
1980 1985 1990 1995 2000 2005 20100.3
0.31
0.32
0.33
0.34
0.35
0.36
0.37
0.38
0.39
0.4
Con
sum
ptio
n V
eloc
ity o
f M3
Euro-area Velocity and Interest Rate
1980 1985 1990 1995 2000 2005 20100
2
4
6
8
10
12
14
16
18
20
Rat
e on
3-m
onth
Eur
ibor
, AP
R
Interest Rate and Velocity Data for Euro Area
1980 1985 1990 1995 2000 2005 20100.2
0.3
0.4
Con
sum
ptio
n V
eloc
ity o
f M3
Euro-area Velocity and Interest Rate
1980 1985 1990 1995 2000 2005 20100
2
4
Rat
e on
Ove
rnig
ht d
epos
its, A
PR
intrest rate
velocity
intrest rate
velocity
velocity
velocity
intrest rate
intrest rate
What To Do, When g, z Are Random?
• Results for Optimal R Completely Unaffected
• Ramsey Principle: Minimize Tax Distortions
– After Bad Shock to Government Constraint:∗ Tax Capital∗ Raise Price Level to Reduce Value of Government Debt
– After Good Shock To Government Budget Constraint∗ Subsidize Capital∗ Reduce Price Level to Reduce Value of Government Debt
91
What To Do, When g, z Are Random? ...
• If there is Staggered Pricing in the Economy, Desirability of Price VolatilityDepends on Two Forces
– Fiscal Force Just Discussed, Which Implies the Price Level Should BeVolatile
– Relative Price Dispersion Considerations Which Suggest that Prices ShouldNot Be Volatile
• Schmitt-Grohe/Uribe and Henry Siu Find:
– For Shocks of the Size of Business Cycles, the Relative Price DispersionConsiderations Dominate
• Henry Siu Finds:– For War-Size Shocks, Fiscal Considerations Dominate.– Some Evidence for this in the Data
96
18001805
18101815
18201825
18301835
18401845
18501855
18601865
18701875
18801885
18901895
19001905
19101915
19201925
19301935
19401945
1950
20
40
60
80
20
40
60
80CONSUMER PRICE INDEX*
War of 1812
Civil War
World War I
WorldWar II
* Base index from 1800 to 1947 is 1967 = 100. Source: US Department of Commerce, Bureau of the Census, Historical Statistics of the US.
yardeni.com
#1
Wars are inflationary. Peace times are deflationary. Historically, prices soared during wars, plunged during peace times. Wars are trade barriers. There is more competition and technological innovation during peace times.
48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00 0220
40
60
80
100
120
140
160
180
20
40
60
80
100
120
140
160
180
B AugCONSUMER PRICE INDEX(1982-1984=100, ratio scale)
B = Fall of Berlin Wall.
yardeni.com
#2
So far, the end of the 50-Year Modern War hasn’t been deflationary globally. Easy money has averted deflation. Nevertheless, inflation is the lowest in 30 years in most industrial economies. There is some deflation in Japan.
- Inflation, War, & Peace -
Deutsche Banc Alex. Brown Best Charts / September 21, 1999 / Page 3
Financing War: Barro versus Ramsey
When War (or Other Large Financing Need) Suddenly Strikes:
• Barro:
– Raise Labor and Other Tax Rates a Small Amount So That When HeldConstant at That Level, Expected Value of War is Financed
– This Minimizes Intertemporal Substitution Distortions
– Involves a Big Increase in Debt in Short Run
– Prediction for Labor Tax Rate: Random Walk.
100
Financing War: Barro versus Ramsey ...
• Ramsey:
– Tax Existing Capital Assets (Human, Physical, etc) For Full Amount ofExpected Value of War. Do This at the First Sign of War.
– This Minimizes Intertemporal and Intratemporal Distortions (Don’t ChangeTax Rates on Income at all).
– Reduce Outstanding Debt
– Make Essentially No Change Ever to Labor Tax Rate
101
Financing War: Barro versus Ramsey ...
– Example:
∗ Suppose War is Expected to Last Two Periods, Cost: $1 Per Period
∗ Suppose Gross Rate of Interest is 1.05 (i.e., 5%)
∗ Tax Capital 1 + 1/1.05 = 1.95 Right Away.
∗ Debt Falls $0.95 in Period When War Strikes.
– Involves a Reduction of Outstanding Debt in Short Run.
– Prediction for Labor Tax Rate: Roughly Constant.
102
A Computational Issue
• Conditional On a Value for λ, Finding Ramsey Allocations Easy (Can UseSimple Linearization Procedures!)
• Policies Can Then Be Computed From Ramsey Allocations.
– Example: Labor Tax Rate Can Be Computed from Ramsey Allocations BySolving for τ t :
ul (ct, lt) + uc (ct, lt)× fn (kt, lt)× (1− τ t) = 0
• But, How To Get λ?
– Get it the Hard Way, Outlined Above
– Under Very Limited Conditions, can Calibrate λ
103
Calibrating the Multiplier, λ
• Conditional on λ :
– Nonstochastic Steady State Consumption, Capital Stock, Labor, Labor TaxRate Functions of λ :
c = c (λ) , l = l (λ)
– Steady State Policy Variable (debt, labor tax, capital tax rate) Can BeComputed:
τ (λ) = 1 +ul (c, l)
uc (c, l) fn (k, l)
• In Practice, τ (λ) is a Monotone Function of λ. Choose λ So That
τ = τ³λ´, τ ∼ Sample Average of Labor Tax Rate
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Problem With Calibrating Multiplier
• Implicitly, this Assumes the Economy Was in an Optimal Policy Regime in theHistorical Sample
• Problem
– When People Compute Optimal Policy, they Want to be Open to thePossibility that Policy Outcomes are Not Optimal
– Want to Use the Ramsey-Optimal Policies as a Basis For RecommendingBetter Policies
• Still, Calibration of λ Works for an Analyst Who Seriously Entertains theHypothesis that Policy in the Sample Was Optimal
• Related to Woodford’s Idea of the Timeless Perspective
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Optimal Monetary Policy When the IntertemporalBudget Constraint Does Not Bind
• Current Generation of Monetary Models Put Government Budget Constraint inBackground by Assuming Presence of Lump Sum Taxes to Balance Budget.
• Ramsey Optimal Policies in These Models Easy to Compute.
107
Optimal Monetary Policy When the Intertemporal Budget Constraint Does Not Bind ...
• Suppose:
– You Have a Very Simple Model, With One Equation Characterizing theEquilibrium of the Private Economy, and One For the Policy Rule.
– The Private Economy Equation is:
πt − βπt+1 − γyt = 0, t = 0, 1, ... (1)
– You Want to Do Optimal Policy. So You Threw Away the Policy Rule.
– The Setup At this Point Has One Equation, (1) in Two Unknowns, πt, yt.Need More Equations!
– The Additional Equations Come In When We Optimize.
108
Optimal Monetary Policy When the Intertemporal Budget Constraint Does Not Bind ...
– Lagrangian Problem:
max{πt,yt;t=0,1,...}
∞Xt=0
βt{u (πt, yt) + λt [πt − βπt+1 − γyt]}
– Equations that Characterize the Optimum: (1), and
uπ (πt, yt) + λt − βλt−1 = 0
uy(πt, yt)− γλt = 0, t = 0, 1, ...
– We Made Up for the One Missing Equation, By Adding Two Equationsand One New Unknown.
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